The Axiomatic Approach to Non-Classical Model Theory

Institution theory represents the fully axiomatic approach to model theory in which all components of logical systems are treated fully abstractly by reliance on category theory. Here, we survey some developments over the last decade or so concerning the institution theoretic approach to non-classical aspects of model theory. Our focus will be on many-valued truth and on models with states, which are addressed by the two extensions of ordinary institution theory known as r-institutions and stratified institutions, respectively. The discussion will include relevant concepts, techniques, and results from these two areas.